Variable Selection Of Quantile Reg-Ression Models With Nonignorable Missing Data

How to make effective statistical inference with nonignorrable missing data is a very challenging problem,especially when the dimension of covariates is very high.In this paper,we propose using a class of weighting approaches with some variable selection procedures for parameter estimation of quantile regression models with nonignorable missing data.Quantile regression is a method for estimating the functional relationship between variables at various positions in the probability distribution.In recent years,it has been widely used in economics and other fields.Various variable selection methods have been well developed to choose the important input variable and to simplify the model under study.Based on the existing theories developed in the context of full observations,we develop two variable selection methods: penalized quantile regression estimation based on the inverse probability weighting approach with and without auxiliary information.Under some regularity assumptions,we prove that the one-step sparse estimator of the inverse probability weighted quantile regression using the SCAD penalty enjoys the oracle property.Due to the non-differentiability of quantile regression and the non-convexity of SCAD penalty function,optimizing the proposed objective functions is not easy.To overcome the computational difficulties,we use a linear programming algorithm to optimize the proposed objective functions.We conduct extensive simulation studies to evaluate the finite sample performance of the proposed method.The simulation results show that all the proposed variable selection methods are reliable under all considered scenarios.By incorporating the auxiliary information,the empirical likelihood based weighting approach produces more efficient estimator.An empirical analysis using the air quality data is also conducted for illustration.